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Area and perimeter. Every Robbins pentagon may be scaled so that its sides and area are integers. ... (2008), "Cyclic polygons with rational sides and area", ...
A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common. [17]: p. 84
That is, a regular polygon is a cyclic polygon. ... Of all n-gons with a given perimeter, the one with the largest area is regular. [10] Constructible polygon
Every regular polygon is a cyclic polygon. For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, … are equal, and sides 2, 4, 6, … are equal). [11] A ...
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
The lengths of the sides of a polygon do not in general determine its area. [9] However, if the polygon is simple and cyclic then the sides do determine the area. [10] Of all n-gons with given side lengths, the one with the largest area is cyclic. Of all n-gons with a given perimeter, the one with the largest area is regular (and therefore ...
A cyclic polygon (one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. [35]
The area of any regular polygon is: = where P is the perimeter of the polygon, and r is the inradius (equivalently the ... The area of a cyclic pentagon, ...